journal of the history of ideas volume 28 issue 3 1967 [doi 10.2307%2f2708626] bruce s. eastwood --...

Grosseteste's "Quantitative" Law of Refraction: A Chapter in the History of Non-Experimental ScienceAuthor(s): Bruce S. EastwoodSource: Journal of the History of Ideas, Vol. 28, No. 3 (Jul. - Sep., 1967), pp. 403-414Published by: University of Pennsylvania PressStable URL: http://www.jstor.org/stable/2708626 .Accessed: 21/06/2014 20:01

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GROSSETESTE'S "QUANTITATIVE" LAW OF REFRACTION: A CHAPTER IN THE HISTORY OF NON-EXPERIMENTAL SCIENCE

BY BRUCE S. EASTWOOD

The first attempt indigenous to the Latin Middle Ages to determine the size of the angle of a refracted light ray was that made by Robert Grosseteste. Composed about 1235, his short treatise on refracted light1 stated that a ray passing from a less dense to a more dense medium will be bent towards the normal so that its path in the second medium is at an angle equal to half the angle of incidence.2 Before considering the exact formulation by Grosseteste of this law, one should become acquainted with its philosophical matrix. In obvious contravention to experimental evidence (later formulated correctly by Snell and Descartes as sin r=k sin i), the half-angle law was Grosseteste's only quantitative determination.

The openly non-experimental nature of this law requires some reflection on his concept of knowledge and its certification. Grosseteste's epistemology lies in a well-defined trend of Neoplatonic metaphysics, which considers interior illumination from a divine source to be the means to true knowledge. This general viewpoint can be found in works of Saint Augustine 3 as well as in later authorities.4 In a form characteristic of its western use in the high Middle Ages, this epistemology is described by Dominicus Gundissalinus, the XIIth-century translator of Avicenna's epistemological work. Gundissalinus' De anima discusses two powers of the rational soul, the contemplative and the active powers, which are concerned respectively with sapientia and scientia. The contemplative intellect is thereby concerned with ultimate truth, while the active intellect obtains knowledge externally, and therefore provisionally.5 This viewpoint goes on to suggest that sapientia is knowledge obtained more directly from a divine intelligence; scientia is contingent knowledge-knowledge based on sensual sources. Both Augustine and Gundissalinus 7 consider the

1 De iride, the treatise in which Grosseteste gives an explanation of the rainbow based on refraction, has been dated to ca. 1235 in Richard C. Dales, "Robert Grosseteste's Scientific Works," Isis, LII (1961), 402.

2 Die philosophischen Werke des Robert Grosseteste, Bischofs von Lincoln, ed., Ludwig Baur (Miinster, 1912), 74. 3De trinitate, XII, 14; De magistro, 12.

4 Etienne Gilson, "Pourquoi Saint Thomas a critique Saint Augustin," Archives d'histoire doctrinale et litteraire du moyen dge, I (1926-1927), 5-127. 5 Gundissalinus, De anima, 10; J. T. Muckle, ed., "The Treatise De Anima of Dominicus Gundissalinus," Mediceval Studies, II (1940), 86-87, 98. 6 De trinitate, XII, 8; De libero arbitrio, II, 13.

7 De anima, 10 (Muckle, "De Anima," 101-102). 403


404 BRUCE S. EASTWOOD

reason for the sensual basis of most knowledge to be man's earthly condition, his physical limitations, his original sin. Grosseteste's epistemology has a great affinity to this outlook. According to him the superior part of the human soul, the intelligence, uses no part of the body in its operation, and can receive knowledge of all things without corporeal instruments, learning only from the divine illumination from above. The rational soul, attaching itself to corporeal things, is overwhelmed by them, and its higher faculties are blinded; the proper effect of sensible knowledge is to correct this. The senses, animated by the soul, receive impressions and thereby excite the soul. The soul is thus incited to draw images of the sensible objects from itself on the basis of single qualities abstracted from the objects by reason.8 The arrival of the intellect at a conclusion and knowledge is thereby achieved by means of sensible perception, but the actual perception and understanding of this knowledge derive ultimately from the divine light.9 The reason for the necessity of appeal to sensory perception is man's earthly condition.10 Only because of the relative servi- tude of man's soul to earthly concerns is his knowledge obtained indirectly: original sin is the sole reason for a difference between the means of knowledge by the blessed and the means of man's knowing intelligibles.

In Grosseteste's commentary on Aristotle's Posterior Analytics, the psychological process of man's knowing is described. Here Grosseteste affirms that knowledge based on the senses is necessary though not the best.1 The inductive way of Aristotle's Posterior Analytics is the basic way of human knowledge; with this is combined deduction, based on knowledge revealed by induction. This method of induction and deduction had been described before Grosseteste not only by Aristotle but also by Galen, Avicenna, and others.12 The use of induction and deduction was discussed by Grosseteste with specific reference to the determination of causes of events.13 His logic of science involved not only induction and deduction in order to determine the causes of phenomena, but also veri-

Gilson, "Saint Augustin critiqu6," 95-96; L. E. Lynch, "The Doctrine of Divine Ideas and Illumination in Robert Grosseteste," Mediaeval Studies, III (1941), 169; Richard C. Dales, "Robert Grosseteste's Commentarius in Octo Libros Physicorum Aristotelis," Medievalia et Humanistica, XI (1957), 15.

Gilson, "Saint Augustin critique," 98; Grosseteste, In Aristotelis Posteriorum Analyticorum librum, I, 14 (edition of Venice, 1521, ff. 20v-21v).

0 Baur, Werke, 138. 11 Grosseteste, Arstotelis Posteriorum, I, 2 (edition of Venice, 1521, f. 4r). 12 A. C. Crombie, Robert Grosseteste and the Origins of Experimental Science

(Oxford, 1953), 74-81; Alfarabi, De scientiis, 4 (Catalogo de las Ciencias, ed. A. Gonzalez-Palencia [Madrid, 1953], 161); Gundissalinus, De anima, 10 (Muckle, "De Anima," 99).

13 Cf. A. C. Crombie's excellent summary of Grosseteste's methodology in "Gros- seteste's Position in the History of Science," Robert Grosseteste, Scholar and Bishop, ed. D. A. Callus (Oxford, 1955), 98-120.


GROSSETESTE'S LAW OF REFRACTION 405

fication and falsification by experiment.14 Yet we have noted that his quantitative law of refraction was non-experimental. The explanation of this apparent paradox is that Grosseteste looked to sensual, experimental knowledge as a second-rate type of knowledge. If he could base knowledge on non-sensory, non-experimental foundations, he could then obtain knowledge with a greater degree of certitude. This attitude is quite in line with his epistemology; it results in a basically non-experimental science.

Grosseteste's law of refraction should not be expected to conform to the rules of experimental science, if we can find in his account an alternative basis, a priori and thereby more certain. In three consecutive and related paragraphs in his De iride, Grosseteste described (1) his quantitative law of refraction, (2) the principle behind it, and (3) the rule for location of an image in refraction. Each paragraph must be considered separately and in detail. In the first paragraph, containing the law of refraction, he said, The size of the angular declination of the refracted ray from a straight ingress can be visualized as follows. First we conceive of a ray which passes from the eye through the medium of the air and meets a second transparent body; we extend into the second transparency the straight line along which this ray travels, and then from the point at which the ray meets the transparency we draw into the depth of that transparency a line which is perpendicular to its surface. I say then that the path of the ray in the second transparency is along a line dividing equally the angle [my italics] which is formed by the imaginary direct extension of the ray and by the line drawn perpendicularly from the point of incidence of the ray upon the surface of the second transparency into that medium.15 This passage quite unambiguously sets up a simple refraction situation. Two media of differing densities are conceived, and their common surface is a plane. A perpendicular, extending into the denser medium, is constructed on this plane. We are directed to imagine a light ray passing first through the rarer medium and striking the plane surface at the point where that surface is met by the perpendicular. If the path of the ray is continued as a straight line from the rarer into the denser medium, we can then say, according to Grosseteste, that the actual path of the light ray in the second body will bisect an angle; that angle is formed by the perpendicular and the rectilinear extension into the second medium of the ray's path in the first. The exact phrasing of this half-angle law is significant. The use of the first person singular (dico) in the statement of the law suggests that

14 Crombie, Grosseteste, 73-74, gives an example, available to Grosseteste via Algazel's logic, which is a paraphrase of Avicenna's logic (for this information I am indebted to Professor Weinberg of the University of Wisconsin). See F. Ueberweg, Die patristische und scholastische Philosophie, ed., B. Geyer (Stuttgart, 1956), 311; C. Prantl, Geschichte der Logik im Abendlande (Berlin, 1957), II, 368. For Avicenna's statement see Avicenna's Psychology, transl. F. Rahman (Oxford, 1952), 55.

15Baur, Werke, 74.



Grosseteste was completely original in this formulation. No such law exists in any earlier known treatise on optics or natural science in general. In the optical works he uses the first person singular only when giving information or opinion patently his own.l- A second point depending on the exact phrasing of the text is the reason for this quantitative law. The italicized phrase, dividentia per aequalia angulum in the Latin text, is significant. It suggests that the idea of equality is being stressed, for the more normal, mathematical way of stating such a law would be, "the re- sultant angle of refraction is one-half the angle of incidence." The frac- tional quantity, one-half, is not mentioned by Grosseteste, however. In itself this phrasing might be explained away as a peculiarity of expression, but, in the light of the succeeding two paragraphs, its significance is real and crucial.

Reinforcing the first paragraph, the next one lays down the principle upon which the half-angle law is based. The text runs as follows. That the quantity of the angle is so determined in the refraction of a ray, similar experiences show us, by which we know that the angle of reflection of a ray upon a mirror is equal to the angle of incidence. And it is shown to us by this principle of natural philosophy, that every operation of nature is by the most finite, most ordered, shortest and best means possible.l In this passage a subsidiary point, illuminating the central idea, is the word used for reflection. At this juncture refractio appears, but it obviously means reflection, being used in discussing a mirror. Furthermore, not even all the manuscripts of this treatise have the term here.18 Until the mid- XIIIth century no fixed terminology for reflection and refraction was cur- rent in northern Europe. Sometimes the word fractio served for both phenomena alternately. Again, as in the Latin version of Alfarabi, the word fractus might refer only to reflection, or even to a specific type of reflection.19 This plurality of usage seems to reflect an ambiguity in the minds of those using the terms. If one uses the same term for reflection or refraction, does one really consider the two phenomena radically different? While Grosseteste exemplifies this generic usage nowhere else in his optical treatises, the phrasing is not necessarily a slip of the pen. For he goes on to couple reflection and refraction under a unifying principle and according to a single quantitative rule. The principle of economy, stated at the end of the paragraph just quoted, seemed to Grosseteste to supply a key for the discovery of the angle of refraction. In its metaphysical sense, as it is used in this passage by Grosseteste, the principle of economy was familiar

16 Essentially the same description of the law as well as the dico appear in an excellent manuscript of Grosseteste's De iride, Bibl. Marucell. C. 163, ff. 19D-20A; this manuscript was not used by Baur in his edition of the optical works.

17 Baur, Werke, 74-75. 18In Bibl. Marucell C. 163, f. 19D, reflexio is used; the word is not a later

addition to the MS. 19Alfarabi, De scientiis, 3 (cf. Gonzalez-Palencia, ed., Catalogo, 150).


GROSSETESTE S LAW OF REFRACTION 407

to the medievals through Aristotle,20 among others. Nature will never act in vain, he says. Grosseteste agrees, citing the quantitative law of reflection in support, for the most economical path from one point to another by way of reflection occurs when the angles of incidence and reflection are equal. This fact appears to him as a striking certification of the principle of economy. The principle should apply therefore to refraction as well, and this he clearly states. The phenomena of refraction are linked with those of reflection as "similar experiences." Since the principle of economy requires equal angles in reflection, refraction must occur in equal angles. The equal angles are the two halves of the angle formed by the perpendicular and the direct extension into the second medium of the path of the ray in the first medium. This essential similarity between reflection and refraction was conceived most clearly by Grosseteste, but may well have been culled from one of his many predecessors in optical discussions. Among others, Seneca, who had no clear knowledge of refraction as such, may have inspired Grosseteste by assuming the phenomena of refraction to be "ex repercussu" and "simile speculo." 21

The basic identity of the two phenomena is argued and, he feels, con- clusively shown by evidence in the third paragraph. In driving his point home Grosseteste stated, A thing which is seen through many transparent media does not appear to be as it truly is, but seems to be at the conjunction of the departing rays, extended from the eye along a straight line, and of a line drawn from the viewed object perpendicular to the surface nearer the eye of the second transparent medium. Moreover, this is shown to us through that experience, and similar reasons, by which we know that a thing seen in a mirror appears at the juncture of the extended line of sight and of a line drawn perpendicular to the surface of the mirror.22 This paragraph is best understood if we start at its end. The last sentence describes the law for location of an image in reflection, which is the following. Imagine a mirror; for the sake of simplicity, a plane mirror. Now imagine a vision of an object by reflection from this mirror. Where does the object appear to be? Draw from the object a line perpendicular to the plane mirror and continuing on behind it. Then draw the line of vision from eye to object via reflection at equal angles. Next draw beyond the mirror a rectilinear continuation of that part of the line of vision from eye to mirror; this will meet the perpendicular at a point equally far from the mirror as is the viewed object. The location of the image in reflection is thus defined by the intersection of a perpendicular to the mirror from

20 Aristotle, De caelo, I, 4, 271a. 21 Seneca, Quaestiones naturales, I, vii, 1-2; see Paul Oltramare, ed., Questions

naturelles (Paris, 1929), I, 33. 2 Baur, Werke, 75. In passing, it is notable that Grosseteste, while speaking of "many transparencies," does not consider multiple refraction, or the possibility of reciprocal angles.



the viewed object and a direct continuation beyond the reflecting surface of the line of vision from the eye to the mirror.23

Mutatis mutandis, this is also the law for location of an image in refraction. The first sentence of the paragraph just quoted describes that law as follows. Imagine a planar common surface between two media of differing densities. Construct a perpendicular to the plane surface. Where the perpendicular and the plane intersect, let a line of vision (from the upper, rarer medium) strike the surface. Then assume this line of vision, refracted at the surface towards the perpendicular, to extend into the denser medium, eventually meeting a visible object. Where will the object appear to be? The location of the image in refraction is defined by the intersection of a perpendicular to the surface from the viewed object and a direct continuation (beyond the refracting surface) of the line of vision from the eye to the surface. While the actual locations of lines in the diagram for refraction would differ from the locations of lines in a reflection diagram, the principle is the same, as the reader will see by comparing the phrasing of the two rules as stated. Finally, note the way Grosseteste passes from refraction to reflection in the paragraph. First refraction is mentioned, and then it is claimed that the same reasoning in detail applies in reflection. In the case of image location, Grosseteste's analogy between the two is quite correct. In determining the size of the angle, the analogy is wrong. But he did not know this. The analogy between the two classes of phenomenon is of basic significance in the development of his optics. Grosseteste considers the analogy to be real and justifies it through the distinction of active and passive aspects of optical occurrences. Light itself is active and acts always in the same way; a mirror or transparent medium is passive and responsible for the differences between reflection and refraction. Although the effects vary, they are only accidental to light. Essen- tially light is always the same. There should be some analogy between the effects in reflection and refraction, for the active principle is more significant than the passive, and the agent (light) is invariant; only the recipient changes. In both reflection and refraction, incident light strikes a surface. Since the agent is the same, the effects should be essentially the same, varying only in their accidents.

The painfully obvious fact that Grosseteste's quantitative law can have no empirical basis has resulted in an interesting pair of modern assessments of the three paragraphs under consideration. In his book on Grosseteste and scientific methodology, A. C. Crombie gives the following translation of the second and third paragraphs to illustrate the quantitative law of refraction conceived by Grosseteste. That the size of the angle in the refraction of a ray may be thus determined, experiments show us similar to those by which we can discover that the reflection of the ray falling on a mirror makes an angle equal to the angle

23 This rule was easily available to Grosseteste in the Euclidean Catoptrics, 16, 17, 18, and in Alkindi's De aspectibus, 21; A. A. Bjornbo and S. Vogl, edd., Alkindi, Tideus, und Pseudo-Euklid, drei optische Werke (Berlin, 1912).


GROSSETESTE S LAW OF REFRACTION 409

of incidence. And this the following principle of natural philosophy shows us, namely that "every operation of nature is the most complete, orderly, brief and best way possible." A thing that is seen through the medium of several transparent bodies does not appear to be as it truly is, but appears to be at the conjunction between the ray passing out from the eye in con- tinuous and direct projection, and the line coming from the things seen which falls on the surface of the second transparent body nearer the eye at equal angles on both sides [my italics]. This is shown by the same experiment as, and similar reasoning to, those by which we know that things seen in a mirror appear at the conjunction of directly projected vision and the line coming to the surface of the mirror at an equal angle.24 The italicized portion of this passage is considered by Crombie to state Grosseteste's quantitative law of refraction, that the angle of refraction is half the angle of incidence when light passes from a rarer to a denser medium.25 Because of the initial mention by Grosseteste of the determination of the size of the angle of refraction, Crombie seems to assume that the passage gives a quantitative law of refraction. The result of such an assumption is misinterpretation, for the quantitative law is not in the passage.

In a recent article 2 Colin Turbayne properly criticizes this interpre- tation by Crombie of the passage just given, although Turbayne's criticism is based in part on a gratuitous assumption of Euclidean knowledge of the location of images in refraction.27 Turbayne correctly reinterprets the passage to show that the italicized portion refers to a line perpendicular to the surface from the viewed object; his diagram shows that the law being given by Grosseteste is actually the law for the location of an image in refraction.2 However, the connection made by Turbayne of the initial section of the quotation, referring to the size of the angle of refraction, with the latter part, which gives the law for the location of an image in refraction,29 is used to reach an unjustified conclusion. Turbayne, like Crombie, apparently assumes that the passage given above is concerned with a quantitative law of refraction because of the initial mention by Grosseteste of the determination of the size of the angle of refraction. Whereas Crombie misinterprets the passage to find in it such a law,30 Turbayne is more imaginative, stating,

24 Crombie, Grosseteste, 123 25 This quantitative law was noted by Ludwig Baur, Die Philosophie des Robert

Grosseteste, Bischofs von Lincoln (t 1253), (Miinster, 1917), 117-18; Crombie, Grosseteste, 123, n. 8.

26C. M. Turbayne, "Grosseteste and an Ancient Optical Principle," Isis, L (1959), 467-72.

2 Ibid., 468. No such knowledge is stated or implied in the optical works attributed to Euclid. 28Ibid., 471. 29Ibid., 470-71. 30 Crombie seems to have been looking for evidence of a quantitative approach; it is interesting to note that in a later article he uses but does not correct Tur- bayne's article in the point of significant error. See A. C. Crombie, "Quantification in Medieval Physics," Isis, LII (1961), 148.



. . . the passage does not reveal Grosseteste's law of refraction. But it does show a rule for finding the angle of refraction experimentally; it does show that he understood and accepted The Ancient Principle [a law for finding the location of an image in reflection and refraction]; and it is consistent with the view that he put his theories to the test of experiment.3a Because this analysis is made with the avowed object of buttressing Crom- bie's view of Grosseteste as an experimentalist,32 there seems to be no consideration of the possibility that a quantitative law of refraction was given by Grosseteste. That such a quantitative law can be found, but in the pre- vious paragraph, neglected by the expositions of Crombie and Turbayne, we have already seen. The rule given in the section discussed by Turbayne was neither intended nor used by Grosseteste to determine experimentally the angle of refraction, for the initial statement of the text in controversy between Turbayne and Crombie refers to the prior, not the succeeding paragraph.

Why should Grosseteste have gone so openly against experimental evidence? Long before his day two major optical treatises, those of Ptolemy and Alhazen, had proven the impossibility of a quantitative law like Grosseteste's. In the later XIIIth century we find a Pole, Witelo,33 and an Englishman, John Peckham,34 using these earlier authorities to good ad- vantage. But in the first half of the century these important works were not known in northern Europe.35 Optical knowledge which Grosseteste did

31 Turbayne, "Ancient Principle," loc. cit., 472. 32 It should be recognized, of course, that Crombie's primary purpose is to show

the logical methodology which Grosseteste felt was applicable to science; however, Crombie's enthusiasm causes him to suggest strongly that Grosseteste was actually an experimentalist in his scientific works. Crombie has independently come to the same conclusion as the present author on this point in an interesting but brief reconsideration found in the preface of a reprinting of his book (Grosseteste, second impression, 1962, p. v).

83 Witelo's Perspectiva was written between 1270 and 1278, according to George Sarton, Introduction to the History of Science (Baltimore, 1931), II, 1027.

4 Peckham's Perspectiva communis was probably written between the mid- 1260's and 1279, according to David Lindberg's unpublished doctoral dissertation, "The Perspectiva communis of John Peckham" (Indiana University, 1965), 9-10.

35 Ptolemy's Optics was not known in northern Europe prior to 1250, according to Albert Lejeune, L'Optique de Claude Ptolemee dans la version latine d'apres I'arabe de l'emir Eugene de Sicile (Louvain, 1956), 31.* The date for the initial knowledge of Alhazen's Optics in northern Europe is not as easy to set. According to George Sarton, "The Tradition of the Optics of Ibn al Haitham," Isis, XXIX (1938), 403-406, there is no basis for alleging the availability of the optics in northern Europe before the mid-XIIIth century; cf. A. Mieli, La science arabe et son r6le dans 'evolution scientifique mondiale (Leiden, 1938), 106, n. 2. Lucien Leclerc, Histoire de la Medicine Arabe (Paris, 1876), II, 516, in seeking to support the attribution of the Latin translation of Alhazen's optics to Gerard of Cremona, cited the following statement of Alpetragius (died early XIIIth century): "Nam licet perspectiva Alhacen sit in usu aliquorum sapientum (sic) latinorum." This is not from Alpetragius, but from a fragment of Roger Bacon's Opus Tertium, en-



seem to have at hand was that contained in works such as the Optics and Catoptrics attributed to Euclid,36 Alkindi's De aspectibus,37 and a pseudo- Euclidean De speculis;38 Aristotle's Meteorology,39 while informative for a reflection theory of the rainbow, gave no information on refraction.40 In those optical sources available, and in other sources as well, refraction was treated in no more than cursory fashion and never quantitatively.41

titled "Liber Tertii Alpetragii" in BN ms. lat. 10264, f. 186; v. Pierre Duhem, ed., Un fragment inedit de l'Opus tertium de Roger Bacon (Quaracchi, 1909), p. 75. Marshall Clagett, Archimedes in the Middle Ages (Madison, 1964), I, 669, has argued for the knowledge of Alhazen's optics by Jordanus in his De triangulis; this need not be contested in order to uphold what I have said above, for in The Medieval Science of Weights, edd. Ernest A. Moody and Marshall Clagett (Madison, 1952), 121-23, Jordanus' works are dated to the 1230's and 1240's, and Jordanus is placed at Toulouse. That Grosseteste himself did not know Alhazen's major optical work is strongly supported by the direct contradiction in De iride of Alhazen's views on quantitative refraction; for Grosseteste's statement see Baur, Werke, 74, and Baur, Philosophie, 117-18; cf. Opticae thesaurus Alhazeni Arabis libri septem, ed. F. Risner (Basel, 1572), bk. VII, chs. 7, 10, and E. Wiedemann, "tber die Brechung des Lichtes in Kugeln nach Ibn al Haitam und Kamal al Din al Farisi," Sitzungs- berichte der Physikalisch-medizinischen Sozietdt in Erlangen, XLII (1910), 16, 33. My point here is not to deny any western knowledge of more advanced optics before exactly 1250, but rather to indicate that such knowledge was acquired closer to 1250 than 1200 and that it spread from the south, reaching Grosseteste at Oxford later than Jordanus at Toulouse.

36 C. H. Haskins, Studies in the History of Mediceval Science (Cambridge, 1924), 178-79, shows these to have been translated from Greek into Latin in Sicily by ca. 1200.

87Translated by Gerard of Cremona (d. 1187); Sarton, Introduction, I, 560. 88Translated by Gerard of Cremona or one of his school; Bjornbo and Vogl,

Optische Werke, 157-158. 9 The first three books were translated from Arabic by Gerard of Cremona; the fourth book was translated from the Greek by Henricus Aristippus (d. 1162); F. H. Fobes, "Medieval Versions of Aristotle's Meteorology," Classical Philology, X (1915), 299.

40 According to E.-M. Antoniadi, "La m6teorologie en grece antique," Bulletin de la Societe Astronomique de France, XLV (1931), 375, Aristotle had knowledge of astronomical refraction, but his citation from Meteorology, III, 2 does not support this; in the indicated passage Aristotle definitely speaks only of reflection. There is the possibility that Aristotle's &vadXaa0v may have been translated by fractio as well as reflectio, though only the latter translation would be correct in Aristotle's usage. It does not appear that Grosseteste misunderstood him.

41See, for instance, Alhazen's shorter treatise on light in J. Baarmann, "Ab- handlung fiber das Licht von Ibn al Haitam," Zeitschrift der Deutschen Morgen- ldndischen Gesellschaft, XXXVI (1882), 224; Euclid, Catoptrics, Def. 6; Bartholomew the Englishman, De proprietatibus rerum, VIII, 42, 43; the pseudo- Euclidean De speculis, 14; the pseudo-Aristotelian De proprietatibus elementorum (Vat. Ms. lat. 2083, f. 209r); Pliny, Historia naturalis, XXXVI, 26; Seneca, Quaestiones naturales, I, vi, 6; Aristotle, Posterior Analytics, I, 31, 88a.



Grosseteste was probably the first to suggest a quantitative law of refraction in the medieval Latin West; his formulation was of necessity inde- pendent of those attempts made by Ptolemy and Alhazen.

Having noted the independence from earlier attempts of Grosseteste's law, the phrasing of the law itself, and his epistemology, we are now in a position to assess the general rationale behind the half-angle law of refraction. His epistemology shows us that he prefers to find a priori principles, if possible, rather than experimental evidence. In the case under consideration the metaphysical principles of uniformity and economy are the guid- ing principles. The principle of economy was given in a metaphysical sense by Aristotle,42 who said that nature creates nothing without purpose. In the second of the three paragraphs under consideration, the principle is enunciated. According to this principle all natural actions are accomplished by the most economical means. The principle of uniformity, as used by Grosseteste, states that a phenomenon always occurs in the same way.43 These two principles are basic to the geometrical optics of Robert Grosse- teste. The principle of economy he states and uses in many places aside from the text being discussed, e.g., in De lineis where he says, ". . nature acts in the shortest and best possible manner." 44 He applies this principle as a proof of the law of reflection, that equal angles are formed by the rays of incidence and reflection, since this is the most economical path by means of reflection between two points.46 Diagrammatically, the two situations, refraction and reflection, are quite obviously different. There is no immedi- ately apparent analogy between the two, because in one case the ray pene- trates a second medium, while in the other case no permeation of the second medium occurs. But Grosseteste had only elementary knowledge in optics, and the disparity between the two phenomena as we today see them was not so radical to him. Having no good reason to assume otherwise, he developed optical rules on the assumption that an underlying similarity existed. Indeed, he had good reason in support of such an assumption; the rule for locating an image in reflection is applicable to refraction as well.

Just as there is a uniformity in the laws for finding the image in reflection and refraction, i.e., at the intersection of a perpendicular to the surface and the continuation of the incident ray, so there should be a uniformity between the quantitative law of reflection and the quantitative law of refraction to determine the most economical path in each case. In reflection the shortest distance between two points by way of reflection is found when the angles of incidence and reflection are equal. What then

42 De caelo, I, 4, 271a (cit. supra, n. 20). 43 Baur, Werke, 60. 44 Ibid, 61. 45 Ibid., 62. It was the "Liber Ptolomei de speculis" (part of Hero's Catoptrics)

which first stated the angles of incidence and reflection to be equal, because this is the shortest path, and light takes the fastest and shortest path. E. Wiedemann, "Zur Geschichte der Lehre vom Sehen," Annalen der Physik und Chemie (neue Folge) XXXIX (1890), 470; translation of the pertinent passage in M. R. Cohen and I. E. Drabkin, A Source Book in Greek Science, 2nd ed. (Cambridge, Mass., 1958), 264.



should be the relationship of the angle of incidence to that of refraction? The principle of uniformity in nature states that a natural action will always occur in the same way, assuming identical conditions; in fact, the agent considered by itself will always act in the same way, but the effects may vary with change in external conditions. This Grosseteste said in De lineis.4 He saw reflection and refraction as two different effects of the same natural action. When light meets a mirror it is reflected. When a transparent medium is placed in the path of a light ray it is refracted. The conditions are varied but the action is the same, for the light ray always acts in the same way. Therefore we can learn, says Grosseteste in the text under discussion, the quantitative law of refraction from the law of reflection. If the light ray always takes the shortest path, and this is defined by an equality of the angles of reflection and incidence in one case, then we can expect refraction to occur with an equality of angles. The equal angles are given by the bisection of the angle formed by the line perpendicular to the surface and the line continuing the path of the incident ray. Grosseteste does not speak of the angle of refraction in terms of the angle of incidence; rather he speaks of the ray which follows the path of a line dividing equally (dividentia per aequalia) the angle between the continuation of the incident ray and the normal. The so-called quantitative law of refraction is therefore only semi-quantitative, for it is based on qualitative principles and might better be called the qualitative law of refraction.

This qualitative approach to light by Grosseteste is not an isolated example. Avicenna treated light itself as quality.47 The qualitative view of light is common in the Neoplatonic context, and pervades other realms of medieval thought as well. In the paintings of Giotto and Fra Angelico, for instance, light is treated qualitatively,48 and the emphasis on light in the Gothic cathedral is essentially qualitative.49 In both cases the qualitative treatment depends on Neoplatonic emphasis on the essential nature of light. The illuminating nature of light, apparent generally in later Ren- aissance art, is evident in the use of shadow, diminution by distance, and such quantitative aspects. The angular law of refraction proposed by Grosseteste is not, however, just qualitative. Equality connotes quantity as well as quality. Quantitatively, the law does say that the angle of refraction is half the angle of incidence. But which aspect of the law is more important to Grosseteste? We suggest that for Grosseteste the qualitative was primary. An interesting parallel exists in the ancient mythopoeic frame of reference.50 While the early Egyptian was quite aware of the length of

46 Baur, Werke, 60 (cit. supra, n. 43). 47 Avicenna, De anima, III, 3 (edition of Venice, 1574, f. 16 va). 48 Wolfgang Schoene, Uber das Licht in der Malerei (Berlin, 1954). 49 Otto von Simson, The Gothic Cathedral (New York, 1962), esp. chs. 1-2.

o5 Henri Frankfort et al., The Intellectual Adventure of Ancient Man (Chicago, 1946), esp. ch. 1 (in Pelican books under the title Before Philosophy); cf. Mircea Eliade, Cosmos and History (New York, 1959), chs. 1-2.



a year, its qualitative aspects were his greatest concern. Also, in dividing the night up into twelve equal parts, the ancient Egyptian astronomers were not giving quantitative parts, for every night of the year had twelve parts.51 Obviously the duration of a part varied through the year. Other examples of mixed qualitative-quantitative outlooks can be given. The point to be made is that an apparently quantitative approach may be qualitative as well. The notion of equality is itself peculiarly susceptible to both approaches. While Grosseteste was not unaware of the quantitative meaning of his law-he even uses the word "quantitative"--the qualitative significance seems to have been uppermost in his mind. Based on qualitative principles, and using the notion of equality in an ambiguous (qualitative-quantitative) way, his angular law of refraction is best understood as a qualitative law.

Ithaca College. 51 Otto Neugebauer, The Exact Sciences in Antiquity (Providence, 1957), 82-86.


Article Contentsp. 403p. 404p. 405p. 406p. 407p. 408p. 409p. 410p. 411p. 412p. 413p. 414

Issue Table of ContentsJournal of the History of Ideas, Vol. 28, No. 3 (Jul. - Sep., 1967), pp. 307-457Volume InformationFront MatterPlato's Treatment of the Theme of the Good Life and his Criticism of the Spartan Ideal [pp. 307 - 324]Newton and the Cyclical Cosmos: Providence and the Mechanical Philosophy [pp. 325 - 346]Leibniz and Locke on "First Truths" [pp. 347 - 366]Dr. John Brown 1735-88) and Early German Romanticism [pp. 367 - 382]NotesPeter Gay and the Heavenly City [pp. 383 - 402]Grosseteste's "Quantitative" Law of Refraction: A Chapter in the History of Non-Experimental Science [pp. 403 - 414]The Spread of Ibn Khaldn's Ideas on Climate and Culture [pp. 415 - 422]The Frightful Consequences of Onanism: Notes on the History of a Delusion [pp. 423 - 431]Godwin's Letter to Ogilvie, Friend of Jefferson, and the Federalist Propaganda [pp. 432 - 444]The Uppsala School and the New Logic [pp. 445 - 450]

ReviewBoas on the Cult of Childhood [pp. 451 - 454]

Books Received [pp. 455 - 456]Back Matter [pp. 457 - 457]

journal of the history of ideas volume 28 issue 3 1967 [doi 10.2307%2f2708626] bruce s. eastwood --...

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